Torsion of Non-Circular Shafts:

In this method, we make supposition to simplify the problem. These suppositions are made in respect of stress components and displacements. These assumptions make governing equations so simple that they may be solved without much difficulty. The assumptions reduce the problem to solving out only one differential equation whereas in a general problem one can require to solve as various as size differential equations. The solving of the problem depends upon assuming the expressions for displacements in three coordinate direction, converting the displacement into strains and then utilizing Hooke's law to find expressions for stress. At last, the stresses are placed in differential equations for equilibrium which are needed to be solved. The assumption of a stress function such that boundary conditions in terms of stress are satisfied is helpful in solving out the equations of equilibrium. The simplifying assumptions in semi-inverse method result in only one differential equation, which is

∂²φ/∂x² + ∂²φ / dy²      =- 2 G θ

Here, θ is the angle of twist and φ is the stress function. We have not prepared ourselves to talk of stress function φ plus to consider the displacement and equilibrium equation. The subject falls under the broader subject of Theory of elasticity. Thus, at this point of time we satisfy ourselves by accepting the results get from semi-inverse method for some well defined non-circular sections.